226 



JV.4 TURE 



[Jan. 6, 1 88 1 



Mathematical Society. His honours of membership were 

 numerous, and are printed on the title-pages of his works. 

 The Pascal-Newton controversy has already been alluded 

 to in these pages, and we -willingly leave it here un- 

 touched. 



" M. Chasles a poursuivi son oeuvre sans interruption 

 depuis sa sortie du Lycce jusqu'a I'age de quatre-vingt- 

 sept ans. Soixante-huit anndes sc'parent la premiere note 

 de Iclcve Chasles, ins(frde dans la CorrcspondiDicc siir 

 P Ecole Polytcclinique, du dernier mcmoire pr^sentc a 

 TAcaddmie des Sciences. Tons les gcomttres, sans dis- 

 tinction de nationalite ni d'ecole, se sont inclines devant 

 ce v(fndrable vieillard ; tous ont admire sa puissance 

 d'invention, sa fdounditd, que I'age semblait rajeunir, son 

 ardeur, et son zele, continues jusqu'aux derniers jours.'' 



A mere recital of the titles of M. Chasles' numerous 

 papers would fill several columns. In the " Catalogue of 

 Scientific Papers" will be found the titles of 177, and 

 from the slight examination we have been able to make 

 we have little doubt that the number published sin:e 

 1873 would bring the total to nearly 240. The subje:ts 

 range over curves and surfaces of the second and of any 

 degree, geometry, mechanics (and attractions), histor)-, 

 and astronomy. Amongst hii earliest papers are those 

 which were translated by the present Bishop of Limerick 

 in 1S41, under the title "Two Geometrical Memoirs on 

 the General Properties of Cones of the Second Degree, 

 and on the Spherical Conies." " These possess strong 

 claims on the attention of mathematicians, whether thcv 

 are considered merely as exercises of pure geomctr\, 

 exhibiting its elegance and power in a remarkable degree, 

 or as a rich and early contribution to the theory of 

 spherical curves." 



Chasles himself remarks in his Rapport^ (which 

 perhaps furnishes the best key to his writings), " On 

 peut s'etonner que, juaque vers la fin du premier tiers de 

 ce siccle, on n'ait eu I'lde'e d'ctudier ni les propridtcs des 

 cones du second ordre qui servent a engendrer les 

 coniques, ni celles des courbes qui tiennent sur la sphere 

 le rang des coniques sur le plan" (p. 75). 



In reply to the question, " On demande un examen 

 philosophique des diffdrentes mdthodes employees dans 

 la geomdtrie rdcente et particulierement de la mdthode 

 des polaires reciproques," was written, " iVIemoire de 

 Geometrie sur deu\ Principes gdndraux de la Science, 

 la Diialite, et I'Homographie" (January, 1830, to the 

 Acaddmie Royale of Brussels), preceded by some histori- 

 cal researches. This work subsequently took the form 

 of the famous "Apergu historique sur I'Origine et le 

 Developpement des Mdthodes en Gdometrie .... suivi 

 d'un I\Iemoire . . . sur deux Principes gendraux . . . 

 et I'Homographie." This work appeared in 1837, and 

 having become exceedingly scarce, was reprinted ver- 

 batim in 1875, with the addition of a shore preface giving 

 a brief historical account of the book. In the Rapport 

 (p. 80) we are told "c'est cette troisieme partie" (the 

 memoir on Duality and Homography) "qui h. donnd lieu 

 ci la composition de I'ouvrige. La thdorie des figures 

 homologiques et celle des polaires reciproques qui sont 

 la baf.e des beau>c travaux de I'illustre Gdndral Poncelet 

 donneient une heureuse impulsion aux recherches de pure 

 geometrie." These two methods were susceptible, he 

 says, of generalisation, and the progress of the science 

 demanded it. The Aperqii, which has been translated 

 into German (except the third part) by Sohncke, is a 

 perfect mine of geometrical facts, and is to the present 

 day a high authority on the subject of which it treats. 

 In some places too great reliance on Montucla (see Dr. 

 AUman on "Greek Geometry from Thales to Euclid," 

 p. 171, cf. also p. 202), and in others non-acquaintance 

 with German ("nous dprouvons un vif regret de ne pouvoir 

 citer ici leurs ouvrages, qui nous sont inconnues, par 



_ ' Pp. 72-126, 220-280, contain 

 tions to geometrj-. 



suite de notre ignorance de ia langue dans laquelle ils 

 sont dcrits,"p. 215) may slightly detract from its merits, 

 but after all deductions it exhibits a vast amount of 

 research and originality, and well merits the title of 

 mivragc classique} 



The appointment to the Chair of Modern Geometry 

 necessitated a course (or courses) of lectures, and in 1852 

 these were embodied in the " Traitd de Gdometrie supdri- 

 eure," "an elaborate and masterly treatise," which of 

 late years has been rarely attainable, and only at a 

 very- high price. M. Chasles, hardly two months before 

 his death, had the satisfaction of seeing a second edition, 

 accompanying which is (pp. 547-585) the excellent " Dis- 

 cours d' Inauguration" (referred to above). The three 

 fundamental principles of this work are "Anharmonic 

 Ratio of Four Points," " nomographic Divisions," and 

 "Involution" {Rapport, \>. 220). 



In 1865 appeared the first volume of the "Traitd des 

 Sections coniques, faisant suite au Traitd de Gdometrie 

 supdrieure." As its title indicates, constant application 

 is made in it of the principles of pure geometry unfolded 

 in the earlier work. It thus differs considerably not only 

 from analytical treatises, but from geometrical treatises 

 also : " Ces trois thdories primordiales s'appliquent avec 

 une e.xtreme facilitd a toutes les recherches concemant 

 les sections coniques" {Rapport, pp. 266 9). 



Mathematicians have long looked for a second volume, 

 materials for which have appeared in the Coinptcs rendiis. 

 In the Rapport (pp. 257-266) will be found an account of 

 the met'^od of geometrical substitution and a definition 

 of the elements (or characteristics) of a system of conies 

 {Coinptcs rendtis, 1864-7). Numerous applications are 

 made of this remarkable theory (for further accounts the 

 English student may refer to Dr. Salmon's " Higher Plane 

 Curves," pp. 360, &c., and "Conies," p. 368; see also 

 later papers in the Coniptcs rcndiis, vol. Ixxviii." p. 577, 

 lie, vol. Ixxw. p. 362, pp. 460-6). 



We must now go back to the year 1863, when Chasles 

 published his " Les trois Livres de Porismes d'Euclide, 

 rdtablis pour la premiere Fois, d'apres la Notice et les 

 Lemmes de Pappus, et conformdment au Sentiment de 

 R. Simson, sur la Forme des Enonces de ces Propositions." 

 In 1838 he had contributed a paper, "Sur la Doctrine des 

 Porismes d'Eu;lide," to Quetelet's Corrcsp. Math. x. (pp. 

 1-23). We must content ourselves with referring to the 

 Rapiport, pp. 155, 233-42 ; the Apcri^H, pp. 39, &;c. (He 

 cites Montucla as to the profoundness of ttie Porisms, 

 gives high praise to Simson, and shows that there is in 

 Pappus's Lemmas what is in effect the projective pro- 

 perty of the anharmonic ratio of four points). The publi- 

 cation of this work led to a short controversy with M. P. 

 Breton (" Question des Porismes — notices sur les ddbats 

 de prioritd auxquels a donnd lieu I'ouvrage de M. Chasles 

 sur les porismes d'Euclide," Paris, 1865; and a second 

 part, Paris, 1S66). M. Chasles comments on these in 

 the Rapport (cf. refif. above). 



We turn now for a moment to the subject of attraction. 

 " La question de I'attractioa presenta-t-elle a I'auteur sous 

 plusieurs points de vue, qui donnerent lieu a divers 

 mdmoires et s'dtendirent meme au prubleme gdndral de 

 r attraction d'un corps de forme queiconque" {Rapport, 

 p. loi); on p. 103 he gives a history of Maclaurin's 

 theorem (of w'hich Todhunter — " History of the Theories 

 of Attraction," S;c., vol. i. 260, writes : " Chasles is 

 correct") ; on p. 105 we read: " Mais il restait toujours 

 it ddsirer une demonstration directe et rigoureuse du 

 theoreme de Maclaurin ; " and he cites an extract 

 from Poinsot's report on his paper {Mhnoires par divers 

 Savants, t. ix. 1846): " Ce mdmoire remarquable nous 

 ofl're un nouvel exemple de Tdldgance el de la clartd que 

 la gdomdtrie peut rdpandre sur les questions les plus 



* De Morgan says, " A work of great importance in the historical point of 



^ " Considerations sur le caractcre propre du principe de correspondance," 

 " S'applique avec une tres grande facilite, a une infinite de questions." 



